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<h1 id="一、熵编码"><a href="#一、熵编码" class="headerlink" title="一、熵编码"></a><font size=3>一、熵编码</font></h1><h2 id="1-什么是熵？"><a href="#1-什么是熵？" class="headerlink" title="1. 什么是熵？"></a><font size=3>1. 什么是熵？</font></h2><p>熵在热力学中，是表示分子状态混乱程度的物理量，这时的熵称为热熵。后来信息论之父香农（C. E. Shannon）把“熵”这一词引入到信息论中，称为“信息熵”，信息越是随机，它的熵值越高。信息熵也是我们在h264这样的数字图像编码中使用的概念。因为我们待编码的图像像素信息、码流的各个句法元素值，其实都属于信息。</p>
<p>而信息熵，就是为了解决信息的量化度量问题，它描述了整个信源的平均信息量。信息熵在我们的熵编码中，表示了信源无损编码后平均码长的下限。所以我们上面才说，熵编码就是为了使编码后，码字的平均码长尽量达到熵极限。而且平均码长越接近熵，说明熵编码的压缩效率越高。</p>
<h2 id="2-什么是熵编码？"><a href="#2-什么是熵编码？" class="headerlink" title="2. 什么是熵编码？"></a><font size=3>2. 什么是熵编码？</font></h2><p>熵编码是无损编码的一种方法。该编码方法的宗旨是找到一种编码，使得码字的平均码长达到熵极限。具体实施是，对出现概率较大的符号，取较短的码长，对出现概率较小的符号取较大的码长。</p>
<h2 id="3-熵编码分类"><a href="#3-熵编码分类" class="headerlink" title="3. 熵编码分类"></a><font size=3>3. 熵编码分类</font></h2><h3 id="3-1-变长编码"><a href="#3-1-变长编码" class="headerlink" title="3.1 变长编码"></a><font size=3>3.1 变长编码</font></h3><p>（1）香农范诺编码</p>
<p>（2）哈夫曼编码</p>
<p>（3）指数哥伦布编码</p>
<p>（4）CAVLC</p>
<h3 id="3-2-算术编码"><a href="#3-2-算术编码" class="headerlink" title="3.2 算术编码"></a><font size=3>3.2 算术编码</font></h3><p>CABAC等。</p>
<h1 id="二、哥伦布编码"><a href="#二、哥伦布编码" class="headerlink" title="二、哥伦布编码"></a><font size=3>二、哥伦布编码</font></h1><h2 id="1-前言"><a href="#1-前言" class="headerlink" title="1. 前言"></a><font size=3>1. 前言</font></h2><p>在计算机中，一般数字的编码都为二进制，但是由于以相等长度来记录不同数字，因此会出现很多的冗余信息，如下：</p>
<table>
<thead>
<tr>
<th align="center">十进制</th>
<th align="center">5</th>
<th align="center">4</th>
<th align="center">255</th>
<th align="center">2</th>
<th align="center">1</th>
</tr>
</thead>
<tbody><tr>
<td align="center">二进制</td>
<td align="center">00000101</td>
<td align="center">00000100</td>
<td align="center">11111111</td>
<td align="center">00000010</td>
<td align="center">00000001</td>
</tr>
<tr>
<td align="center">有效字节</td>
<td align="center">3</td>
<td align="center">3</td>
<td align="center">8</td>
<td align="center">2</td>
<td align="center">1</td>
</tr>
</tbody></table>
<p>如数字1，原本只需要1个bit就能表示的数据，如今需要8个bit来表示，那么其余7个bit就可以看做是冗余数据，在网络传输时，如果以原本等长的编码方式来传输数据，则会出现很大的冗余量，加重网络负担。但是如果只用有效字节来传输上述码流，则会是：101 100 11111111 10 1，这样根本不能分离出原本的数据。</p>
<p>哥伦布编码则是作为一种压缩编码算法，能很有效地对原本的数据进行压缩，并且能很容易地把编码后的码流分离成码字。</p>
<h2 id="2-编码基本思想"><a href="#2-编码基本思想" class="headerlink" title="2. 编码基本思想"></a><font size=3>2. 编码基本思想</font></h2><p>一个码字的信息量，称之为熵，二进制上可用log2[n]来表示，也就是上面表格的有效字节，但是如果只是把有效码字串联起来，得到的只是一串无用的码流，因为这串码流中并没有描述单一码字的信息量，也就是无法对码流进行分离。</p>
<p>哥伦布编码就采用了加0前缀，用于表达码字的信息量，在得到N个0前缀后，就能知道该码字在码流中的长度，并从码流中把码字分离出来。</p>
<h2 id="3-哥伦布码"><a href="#3-哥伦布码" class="headerlink" title="3. 哥伦布码"></a><font size=3>3. 哥伦布码</font></h2><p>哥伦布码就是将编码对象分能成等间隔的若干区间（Group），每个Group有一个索引值：Group Id。对于Group Id采用一元码编码；对于Group内的编码对象采用定长码。如下图：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-01-%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241215094528731.png" alt="image-20241215094528731"  />

<p>对于任一待编码的非负正整数N，Golomb编码将其分为两个部分：所在组的编号GroupID以及分组后余下的部分，GroupID实际是待编码数字N和参数m的商，余下的部分则是其商的余数，具体计算如下：</p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">#对于编码对象n</span><br><span class="line">n = q * m + r = low(n/m) * m + r</span><br><span class="line">其中</span><br><span class="line">	q 是 Group Id，q = low(n/m)，表示取下整; </span><br><span class="line">	r 是 余数，表示 Group 内编码对象。</span><br></pre></td></tr></table></figure>

<blockquote>
<p>Tips：</p>
<p>Golomb编码是一种无损的数据压缩方法，由数学家Solomon W.Golomb在1960年代发明。Golomb编码只能对非负整数进行编码，符号表中的符号出现的概率符合几何分布(Geometric Distribution)时，使用Golomb编码可以取得最优效果，也就是说Golomb编码比较适合小的数字比大的数字出现概率比较高的编码。它使用较短的码长编码较小的数字，较长的码长编码较大的数字。</p>
</blockquote>
<h1 id="三、编码原理"><a href="#三、编码原理" class="headerlink" title="三、编码原理"></a><font size=3>三、编码原理</font></h1><blockquote>
<p>哥伦布编码使用指定的整数 M 把输入的整数分成两部分：商数 q、余数 r。 商数当做一元编码，而余数放在后面做为可缩短的二进制编码。</p>
</blockquote>
<h2 id="1-一元编码"><a href="#1-一元编码" class="headerlink" title="1. 一元编码"></a><font size=3>1. 一元编码</font></h2><p>一元编码(Unary coding)是一种简单的只能对非负整数进行编码的方法，对于任意非负整数num，它的一元编码就是num个1后面紧跟着一个0。例如：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-01-%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241215101836760.png" alt="image-20241215101836760"  />

<p>编解码的伪代码如下：</p>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line">UnaryEncode(n) &#123;</span><br><span class="line">    <span class="keyword">while</span> (n &gt; <span class="number">0</span>) &#123;</span><br><span class="line">        WriteBit(<span class="number">1</span>);</span><br><span class="line">        n--;</span><br><span class="line">    &#125;</span><br><span class="line">    WriteBit(<span class="number">0</span>);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line">UnaryDecode() &#123;</span><br><span class="line">    n = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">while</span> (ReadBit(<span class="number">1</span>) == <span class="number">1</span>) &#123;</span><br><span class="line">        n++;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> n;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h2 id="2-余数处理"><a href="#2-余数处理" class="headerlink" title="2. 余数处理"></a><font size=3>2. 余数处理</font></h2><p>对于余下的部分r则有根据编码数字大小的不同有不同的处理方法。</p>
<ul>
<li>如果参数m是2的次幂（Golomb-Rice编码）</li>
</ul>
<p>使用取r的二进制表示的低log2⁡(m) 位，作为r的码字。</p>
<ul>
<li>如果m不是2的次幂</li>
</ul>
<p>设 b &#x3D; &lceil; log2(m)&rceil;，这里时若 r &lt; 2^b - m，则使用b-1位的二进制编码r；若 r ≥ 2^b - m，则使用b位二进制对 r + 2^b -m 进行编码。</p>
<blockquote>
<p>Tips：</p>
<p>&lceil; a &rceil;表示大于 a 的最小整数的ceil运算。</p>
</blockquote>
<p>例如，m&#x3D;5时，有：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-01-%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216093445607.png" alt="image-20241216093445607"  />

<h2 id="3-总结"><a href="#3-总结" class="headerlink" title="3. 总结"></a><font size=3>3. 总结</font></h2><p>设待编码的非负整数为N，Golomb编码流程如下：</p>
<p>（1）初始化正整参数m；</p>
<p>（2）取得组号q以及余下部分r，计算公式为：q &#x3D; N&#x2F;m，r &#x3D; N%m；</p>
<p>（3）使用一元编码的方式编码q；</p>
<p>（4）使用二进制的方式编码r。如果参数m是2的次幂（Golomb-Rice编码，指数哥伦布编码），使用取r的二进制表示的低log2⁡(m) 位，作为r的码字；如果m不是2的次幂，设 b &#x3D; &lceil; log2(m)&rceil;，若 r &lt; 2^b - m，则使用b-1位的二进制编码r；若 r ≥ 2^b - m，则使用b位二进制对 r + 2^b -m 进行编码。</p>
<h1 id="四、Golomb-Rice-编码"><a href="#四、Golomb-Rice-编码" class="headerlink" title="四、Golomb-Rice 编码"></a><font size=3>四、Golomb-Rice 编码</font></h1><h2 id="1-Golomb-Rice-编码是什么"><a href="#1-Golomb-Rice-编码是什么" class="headerlink" title="1. Golomb-Rice 编码是什么"></a><font size=3>1. Golomb-Rice 编码是什么</font></h2><p>Golomb-Rice是Golomb编码的一个变种，它给Golomb编码的参数m添加了个限制条件：m必须是2的次幂。这样有两个好处：</p>
<p>（1）不需要做模运算即可得到余数r：r &#x3D; N &amp; (m - 1)；</p>
<p>（2）对余数r的编码更简单，只需要取r的二进制的低log2(m)位即可。</p>
<h2 id="2-总结"><a href="#2-总结" class="headerlink" title="2. 总结"></a><font size=3>2. 总结</font></h2><p>Golomb-Rice的编码过程更为简洁：</p>
<p>（1）初始化正整数m，m必须为2的次幂。</p>
<p>（2）计算q和r：q &#x3D; N&#x2F;m，r &#x3D; N &amp; (m - 1)。</p>
<p>（3）使用一元编码编码q。</p>
<p>（4）取r的二进制的低log2(m)作为r的码字。</p>
<h1 id="五、指数哥伦布编码"><a href="#五、指数哥伦布编码" class="headerlink" title="五、指数哥伦布编码"></a><font size=3>五、指数哥伦布编码</font></h1><h2 id="1-什么是指数哥伦布编码？"><a href="#1-什么是指数哥伦布编码？" class="headerlink" title="1. 什么是指数哥伦布编码？"></a><font size=3>1. 什么是指数哥伦布编码？</font></h2><p>Rice的编码方式和Golomb的方法是大同小异的，只是选择m必须为2的次幂。而Exp-Golomb则有了一个很大的改进，不再使用固定大小的分组，而使组的大小呈指数增长。如下图：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-01-%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216094952314.png" alt="image-20241216094952314"  />

<p>Exp-Golomb的码元结构是： </p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">[M zeros prefix] [1] [Offset]</span><br></pre></td></tr></table></figure>

<p>其中M是分组的编号GroupID，1可以看着是分隔符，Offset是组内的偏移量。</p>
<h2 id="2-K阶Exp-Golomb"><a href="#2-K阶Exp-Golomb" class="headerlink" title="2. K阶Exp-Golomb"></a><font size=3>2. K阶Exp-Golomb</font></h2><p>Exp-Golomb需要一个非负整数K作为参数，称之为K阶Exp-Golomb。其中当K &#x3D; 0时，称为0阶Exp-Golomb，目前比较流行的H.264视频编码标准中使用的就是0阶的Exp-Golomb，并且可以将任意的阶数K转为0阶Exp-Golomb编码。</p>
<blockquote>
<p>参考资料：</p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/CrystalShaw/article/details/129388266">h264编码概述八(哥伦布编码ue(v))-CSDN博客</a></p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/easyhao007/article/details/109896846">Exp-Golomb 指数哥伦布编码-CSDN博客</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/TaigaCon/p/3571651.html">指数哥伦布编码 - TaigaComplex - 博客园</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/DwyaneTalk/p/4035206.html">H.264学习笔记6——指数哥伦布编码 - DwyaneTalk - 博客园</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/wangguchangqing/p/6297792.html">Golomb及指数哥伦布编码原理介绍及实现 - Brook_icv - 博客园</a></p>
</blockquote>

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